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Full text of "Optimal fiscal and monetary policy, and economic growth"

AUG 21V 


DEWEY LIBRARY 



Optimal Fiscal and Monetary Policy, 
and Economic Growth 

by 
Duncan K. Foley 
Karl Shell 
Miguel Sidrauski 

Number 21 - April 1968 



t< 



c 



The views expressed in this paper are the authors' 
sole responsibility, and do not reflect those of the 
Department of Economics, nor of the Massachusetts 
Institute of Technology. 



1. Introduction 

There have been two broad strategic approaches to the study of economic 
growth. The first, exemplified by Solow's paper [12], attempts to explain 
how an enterprise economy will grow given its technology and the market be- 
havior of its consumers. The second approach, exemplified by Ramsey's paper 
[7], attempts to determine a optimal development strategy for a fully planned 
economy, given its technological constraints. 

Both of these approaches fail to capture the central policy problem 
of a modern "mixed" economy in which the government can influence investment 
and saving, but only indirectly by manipulating certain basic variables 
like the deficit and the money supply. This paper represents an attempt 
to begin the analysis of this problem. 

The very term "mixed economy" implies that there are two centers of 
decision-making, and that the preferences of the consumers and of the gov- 
ernment are distinguishable. It is not at all clear where the preferences 
of the government come from, or if governments do have consistent prefer- 
ences of the kind we will talk about. But a constant theme of policy lit- 
erature is that government intervention in the economy is effective and 
can be judged as good or bad for the economy without direct reference to 
consumer preferences. This is particularly true of policy prescriptions 
for economic growth. It seems to us that postulating a social welfare 
functional for the government is the best way to make rigorous the pre- 
scription of government control in the mixed economy. 



In a very simple dynamic model, Nelson [3] studied the monetary and fis- 
cal policies consistent with full employment. Using a different model, 
Phelps [ 5 ] has studied the effects of different monetary and fiscal 
policies on the level of investment and the rate of inflation. 



-2- 

We begin with a model of a market economy with two produced commodi- 
ties: consumption goods and investment goods; and three assets: money, 
bonds and capital. From the usual two-sector production model, we derive 
demands for the productive services of capital and labor, which are sup- 
plied inelastically at any moment, and supply flows of consumption and 
investment goods, which depend only on factor endowments and the consump- 
tion price of capital. We assume that the demand for consumption is pro- 
portional to net disposable income which includes government taxes and 
transfers, made in a lump-sum fashion. Given the consumption goods price 
of capital and the nominal value of net government transfers, equilibrium 
in the market for consumption goods can be achieved only at some equili- 
brium consumption goods price of money. 

Given the consumption goods price of money, the markets for stocks 
determine prices and rates of return for three assets: money, bonds, and 
capital. We do not attempt to derive demand functions for assets from 
individual maximizing behavior, but we believe our formulation is fairly 
general. Since the markets for assets and consumption goods must equili- 
brate simultaneously, the consumption goods price of capital, the consump- 
tion goods price of money and the bond interest rate are jointly determined 
in these markets. The price of capital is of particular importance because 
it determines the flow of outputs of consumption and investment goods. 
Producers note the going price of capital and supply as much new investment 
as is profitable for them at that price. The new capital finds a place in 
portfolios through the accumulation of saving and, if necessary, through 
a change in the price and rate of return to all capital, old and new. 

The government has already appeared twice. First, its deficit ap- 
pears as transfer income and influences the demand for consumption goods. 
Second, the government can change the relative supplies to the asset mar- 



-3- 

ket of its own bonds and money by making open market purchases or sales. 
Changes of this kind will affect the equilibrium price of capital, and will, 
therefore, affect the economy's growth path. 

We assume the government has two goals: maximization of the integral 
of discounted utility of per-capita consumption; and the management of 
aggregate demand to achieve a stable consumer price level. We describe 
the optimal growth path for consumption and investment which, for a given 
welfare functional, depends only on the technology and initial capital- 
labor ratio, since these two facts are the only real constraints on pos- 
sible paths. To achieve this path, while maintaining stable consumer 
prices, the government must manipulate its deficit and open market policy 
to induce the private sector to produce investment at the optimal rate at 
each instant. 

If we compare this model with the conventional optimal growth model, 
we see that the private asset demands and consumption behavior are like 
additional constraints from the government's point of view. They are 
facts which the government must operate with to achieve its goals. 

Optimality implies a specific path for the government decision 
variables, particularly the deficit, the stock of government debt, and 
the composition of that debt. The mixed economy with optimal monetary 
and fiscal policy tends to a unique capital-labor ratio and a unique per- 
capita government indebtedness independent of initial endowments. 

For the special case in which the instantaneous utility function 
of per-capita consumption has constant marginal utility, we show that 
the deficit increases with the capital stock along the optimal path. 

We are also able to establish propositions concerning the relation 
between long-run optimal values for the per-capita capital stock, the 
per-capita debt and the composition of the debt for economies which 



-4- 

have different social rates of discount and different private saving pro- 
pensities. It is possible in our model for an economy with a lower social 
rate of discount and thus a higher long-run capital-labor ratio, to have 
a larger long-run per-capita government debt. 

We draw a final conclusion which bears on the idea of the "burden 
of the debt." In this model the initial stock of debt has no effect at 
all on the optimal growth path of consumption and investment. The econ- 
omy's growth possibilities are contrained only by its technolgy and its 
initial endowments of capital and labor. There is no per se burden to 
the debt, although the accumulation of the debt may have been partly at 
the expense of capital accumulation. 

These results represent only the beginnings of a satisfactory theory 
of growth policy in an indirectly controlled market economy. In particu- 
lar, we expect that this analysis can be extended to models which differ 
somewhat in their descriptions of market behavior; for example, to models 
with more general consumption functions. 

2. Production 

Our production model is the simple two-sector constant returns model 
of Uzawa [13] . The heavy curve in Figure I is the production possibility 
frontier (PPF) corresponding to a capital-labor ratio k. Production will 
take place at the point where the PPF has slope ( - P k ) » where p, is the 
price of capital relative to consumption goods. Inspection of the figure 
shows that, where y and y are per-capita output of investment and con- 
sumption 



(2.1) y T = y^k, p fc ) , y Q = y c <k, p k > , 



with k and p, uniquely determining y and y . Also 

K. i. L* 



-5- 



y(k,p k ) 



y c (k,p k ) ■- 




^ slope (-p k ) 



y i (k -V 



i y(k,p k )/p k 7 



Figure I 



-6- 



(2.2) £y T ay 

ap k ap k 



Three other comparative statics propositions will be needed in our 
analysis. For a full discussion of these propositions, see Ry b czinski 
[8] or Uzawa [13]. We assume that production of consumption is always 
more capital-intensive than production of investment. This implies that 



(2.3) 9y ay 

air < °« aiT * °" 



Another important fact is that under the capital intensity assumption, 
the rental rate on capital depends only on the price of capital and 
declines as the price of capital rises. 

(2.4) r = r(p k ), r'(p k ) < 0. 

Further, the capital intensities depend only on p, , and rise as p, rises. 

(2.5) kj = kjCp^, kj > 0, 

k c = k c<Pk>> k i >0 « 

where k is the capital intensity in the i-sector (i = I, C) . 

3. The Asset Market 

We assume that there are three types of assets that can be held in 
the portfolios of wealth owners: physical capital, government non- interest- 
bearing debt called money, and interest-bearing debt called bonds. A unit 
of capital yields a return p, given by 

(3.1) p k = r(p k )/p k + TT k , 

where r is the rental rate on capital and tl is the rate at which individuals 



-7- 



expect p, to change. In the previous section we showed that r is a decreasing 

function of p, . 
k 

We assume money yields no interest payment, so its rate of return p is 
given by 



(3.2) p = TT 
v 'mm 

where tt is the expected rate of change in the consumption goods price of 

money p . 

For simplicity, we assume that bonds (like savings deposits) have 

variable income streams but that their money price remains constant. 

By a proper choice of units, we can set the price of bonds p, = p , and 

d m 

thus the rate of return on bonds is 



(3.3) p = i + TT 

b m 

where i is the bond rate of interest. 

At each moment of time, the real per-capita quantities of capital 
kp, , money mp , and net holdings of government bonds bp , that wealth 
owners desire to hold in their portfolios depend upon their real per- 
capita wealth, a; the rates of return on capital, p,; money, p ; and 

K. IQ 

bonds, p, ; and upon the consumption value of per-capita output, y ■ 
y r + p.y T (representing the transactions motive for holding assets). 
When the asset market is in equilibrium: 

(3.4) kp k = J(a, y, p fc , p^, p fa ) , 

(3.5) mp m - L(a, y, p fc , P m> P b ) , 

(3.6) bp m = H(a, y, p k> Pffi , pj , 
with 

(3.7) a « kp, + (b + m)p m = kp, + gp , 



-8- 

where g is the aggregate per-capita stock of government debt. By Walras' 
Law, it follows that given the budget constraint (3.7), if any two of the 
three asset markets are in equilibrium, the third one must also be in equil- 
ibrium. Therefore, given it, , p , and it , and the supplies of assets, any 
two of equations (3.4) - (3.6) together with (3.7) determine the equilibrium 
price of capital p, and the rate of interest i. We assume that assets are 
gross substitutes for each other, and that none of the assets is inferior, 
i.e., that wealth elasticities of demand are positive. 

We assume that only the government can issue money, so the net per- 
capita holdings of money in private portfolios must be non-negative, i.e., 
m _> 0. On the other hand, the private sector can issue interest-bearing 
debt and the government could choose to be a net holder of bonds, allowing 
b to be negative. In fact, net government bond holdings cculd be sufficiently 
large to cause the government to be a net creditor, when g = b + m < 0. 

Through open market purchases and sales, the government determines 
the composition of its outstanding debt. We call the debt-money ratio 
x ■ g/m. An open market purchase increases the supply of money and leaves 
g unchanged, thereby lowering x when g > 0. 

Proposition 1 : An open market purchase increases the equilibrium price 
of capital p, and lowers the equilibrium rate of interest i. 

Proof : Substituting x = g/m = (b + m)/m in (3.4) and (3.5) and implicitly 
differentiating yields 

(3.8) 3p k -gP m (5J/3p b ) < . > _ 

3T 2~ > ° as s K °- 

x A 



where 



A o (3L/3p k )(3j/3p b ) -I |^- - klf 3L/3p b J 



< 0, 



-9- 



since by assumption 

(3L/8p k ) = k(3L/3a) + (3L/3y) (3y/3p ) + (3L/3p k ) OP k /3p k ) > 0, 



and 



|^- -k = k(|^-l/ + (3J/3y)(3y/3p k ) + (3J/3p k ) OP k /3p k ) < 0. 
Similarly 

|^ < as g < 0. 

3x ° 

An open market purchase forces a change in equilibrium asset prices 
and rates of return. In particular, when it, , p , IT , and k are held constant, 
the rate of interest i will have to fall in induce wealth-owners to hold a 
larger amount of money and a smaller amount of bonds in their portfolios. 
The fall in the rate of interest increases the demand for capital, thus lead- 
ing to an increase in the price of capital. This heuristic argument is for- 
malized in the comparative statics of Proposition 1. 

Next, we make an important assumption. We assume that the demand func- 
tions J(*)» !■(•), and H(«) are sufficiently "flexible" so that given g and 
k > 0, the government by setting the current level of x will be able to set 
the price p, at any level consistent with tangency of the national income 
isoquant with the PPF in Figure I. Thus, we assume that by varying the debt- 
money ratio, the government is able to achieve any efficient mix of consump- 
tion and investment. The most likely difficulty in meeting this requirement 
is that there might be some rate of return to capital so low that increasing 
m will lead to no rise in p.. What we are ruling out here is a kind of 



Alternatively, the assumption is equivalent to saying that by merely varying 
x, the government will be able to trace out all points on the PPF of Figure I. 
The comparative statics and comparative dynamics of the descriptive model 
are treated in greater detail in [2]. 



-10- 



"liquidity trap," since we assume monetary policy is at least potent enough 
to achieve any value of p, for which production is not completely specialized. 

4. Savings and Growth 

The government issues debt in order to finance its budget deficit. 
Let d denote the per-capita government deficit; then 

(4.1) g = d - ng, 

where n is the relative rate of population growth. 

We define per-capita net disposable income y as the value of factor 
payments (equal to the value of output) , net government transfers to the 
private sector, and expected asset appreciation. Since we assume that 
there are no government expenditures, the value of net government trans- 
fers is equal to the government budget deficit, so that 

(4.2) y = y c + Pkyi + Pm d + v + \ k - 

2 
In what follows we simplify by assuming that ir, ■ 0. We also assume that 

ir =0, and this seems to be particularly justified because we will study 
only situations in which the government manipulates fiscal and monetary 
policy to achieve a constant price level and therefore a constant p . 
We further assume that individuals aave a constant fraction s of 
income y, so that if it, = = it , for the commodity market to be in equil- 
ibrium 



(4.3) y c = (1 - s)y = (1 - s) [y + p m d] , 



See [11] , where y is called per-capita Individual Purchasing Power. 

While in the short-run p. may change, in our analysis $. tends to zero 
in the long run, thus lending some justification to our assumption. 



-11- 

where y = y c + P k y z< 

At any moment, k and g are historically given by inherited endowments. 
If the government sets the debt-money ratio at x, then we can think of the 
price of capital p, as being determined in the asset market, i.e., by 
equation (3.4) - (3.7). Given k and p, , producers determine y , y T and 
thus y. 

Then there are two ways in which we can view equation (4.3). If 
the per-capita deficit is d, then equation (4.3) can be solved for the 
price of money p that will clear the commodity market. Alternatively, 
if the government wants to sustain some price level (1/p ), then (4.3) 
can be solved for that per-capita deficit d that will clear the commodity 

market when p is held equal to p . In what follows, we consider the 

r m ^ *m 

case in which the government is commited to pursue a constant-price-level 

policy, i.e., a policy which sustains forever the initial price level 

(l/p°) , so that p = 0. Then from (4.3) 
r m m 



(4.4) 



d = <Kk, P k ). 



Substituting (4.4) in (4.3) and differentiating yields 



(4.5) 



i£ - _i 



8k 



m 



1 - s 9k 



3y c 3y x 

+ Pk 3k~ 



> o, 



by (2.3) and 



(4.6) 



3j) 
9P, 



m 



*y, 



- p 



!l2 

k3p, 



- yi 



< o, 



by (2.2). 

Since p, and k uniquely determine y , capital accumulation is given 
by 






-12- 



(4.7) k = y r (k, p k ) - nk. 

Equation (4.7) can be rewritten as 

sy - (1 - s)dp 

(4.8) k = - nk. 

P k 

5. Optimal Growth in the Fully Controlled Economy 

Suppose that the central planner can directly command the allocation 
of resources. Suppose also that the planner's notion of instantaneous 
welfare is based exclusively on per-capita consumption y . Notably, the 
planner is assumed to take no direct account of the population's asset pref- 
erences. We can assume that the planner seeks to maximize the intertemporal 
welfare functional 

00 

(5.1) / U[y r (t)]e _<5t dt, 

where marginal utility U' is positive but declining; U" < 0. 6 > is the 
planner's pure subjective rate of time discount. Choosing utility as the 
numeraire, socially valued, discounted, per-capita national product H is 
given by 

(5.2) H = [U(y c ) + q(y x - nk)]e" 6t , 

where q is the demand price of investment goods in terms of utility cur- 
rently foregone. For a program to be feasible 

(5.3) k = y T - nk and k(0) = k Q . 

For a program to maximize (5.1) subject to the two-sector technology and 



This section is essentially a review of established results in two-sector 
optimal growth theory. See, especially, Cass [1]. Analysis of the case 
with linear objective functional appears in [14] and [9]. 



-13- 

(5.3), y and y must be chosen at each instant so as to maximize national 
« I 



product H. Thus 



(Hi) 

1 \ dy C/PPF 



D*(y c )+qlTri = °- 



which is equivalent to 

(5.4) U'(y c ) = q/p k . 

First-order condition (5.4) states that the marginal utility of per-capita 

consumption must be set equal to the utility price of investment divided 

1 
by consumption price of investment. 

Along the optimal trajectory, the social return on a unit of capital 

must be equal to the discount rate 6, 



(5.5) q7q +r[k I (p k )] = 6 + 



where k is the efficient capital-labor ratio in the investment goods 

1 2 

industry when the consumption price of capital is p . 

It is further required for optimality that the present discounted, 

utility value of the per-capita stock tend to zero, 

(5.6) lim q(t)e" 6t k(t) = 



In (5.4) it is assumed that on an optimal trajectory the utility demand 
price of investment is equal to the utility supply price of investment. 
This assumption means that the optimal allocation is not completely spe- 
cialized. For the fuller analysis, see [1], [9], and [14]. 

2cf. Pontryagin, et.al. , [6]. The condition is that d(qe~ )/dt = -9H/3k, which 
reduces to (5.5) after applying an envelope theorem to the expression (dy/dk) . 
The population growth rate n appears on the RHS of (5.5) because q is the utility 
price of a unit of k rather than K. The social rate of return to K must equal 6; 
thus the rate of return to k must equal 6 + n. Equation (5.5) has some impli- 
cations for decentralization. If the price system q(t) obtains, factors are 
rewarded by marginal products, and the government sells for a unit of utility 
a consol that pays the instantaneous rate 6, then (5.5) is the perfect-foresight 
market clearing equation. 



-14- 



From (5.5), q = when 

(5.7) rtkj-fej.)] = 6 + n. 

Since we assume that efficient capital-intensities do not cross, we know 
from the two-sector production model that (5.7) holds for a unique consump- 
tion price of investment p *. So that from (5.4), q = if 

q = p k *u'(y c ), 



yielding 
(5.8) 

by (2.3). 



(ajL " *' ^0 & < ■ 



From (5.3), k = when k = y_/n. 

1 
It can be shown that the k = curve crosses the q = curve from the left. 

Their unique intersection is labelled (k*, q*). It also follows that 



lim (q) . = °° and lim (q) . = °° 
k-K) k=0 k*£ k=0 

where K. is the maximum sustainable capital-labor ratio. 

The laws of motion, for the system (5.3) and (5.5) are described in 

the phase diagram of Figure II. The unique stationary point, (k*, q*) , 

is a saddlepoint. The heavy arrows indicate the locus of points (k,q) 

tending to (k*,q*). Trajectories not tending to (k*,q*) can be shown 

to violate (5.6). Therefore, given the initial capital-labor ratio 

k n , q(0) is uniquely chosen by the planner so that [k_, q(0)] lies on the 

See Cass [1]. 



-15- 

heavy line in Figure II. 

Several properties of the optimal solution are noteworthy. 

Proposition 2 : On an optimal path, the long-run marginal product of 
capital is equal to the rate of population growth plus the rate of discount. 

This follows immediately from considering the stationary solution to 
(5.5). Thus, as the rate of discount becomes small, 6 -► 0, the optimal 
long-run marginal product of capital approaches the rate of growth, and 
so the long-run capital-labor ratio approaches the Golden Rule value. 

Proposition 3 ; If the initial capital- labor ratio k_ is less than 
the long-run optimal capital-labor ratio k*, then along the optimal tra- 
jectory: (1) k is increasing through time; and (2) the utility demand 
price of investment q is decreasing through time. If the initial capital- 
labor ratio k n is greater than the long-run optimal capital-labor ratio 
k*, then along the optimal trajectory: (1) k is decreasing; and (2) 
q is increasing. 

Proposition 3 implies that along an optimal trajectory, sign(k) = 
sign(-q) = sign(k*-k). That is, 

(5.9) (dq/dk) < 0. 

Proposition 4 : (1) If the initial capital-labor ratio k_ is 
less than k*, then on the optimal trajectory, per-capita consumption 
is increasing. (2) If the initial capital-labor ratio is greater 
than k*, then on the optimal trajectory, per-capita consumption is 
decreasing. 

Proof : Time differentiation of y yields 

9y c . 3y c /9p k • 3p k • 



(5 - 10) y c = 3ir k + 3irV37~ q + 3k~ k 






-16- 




q* 



Figure II 



-17- 



(5.11) 



8q 



± 

fr + u" 






*"| > 



by (2.2) and 



(5.12) 



'If* 
3k 



3y ( 



Nt + ^] 



> o 



by (2.2) and (2.3). Combining (5.10)-(5.12) yields 



(5.13) y 



C 9k 



p^(u" (yc ))!lc + ^ 

8p k Pk 



9y c 9p k • 

k + 3p^3q~ q 



- as k - k* 



by Proposition 3. 

Proposition 5 ; On an optimal trajectory, (1) if k < k*, then p, < p, *; 
while (2) if k > k*, then p, > p *. 

Proof: p = p * if and only if q = 0. But for k < k*, the optimal 
trajectory lies below the q = curve. Therefore, by (5.11), p, < p* and 
likewise for k > k*. 

Since at k* the optimal consumption price of investment is p,*» an 
immediate corrollary to Proposition 5 is that on an optimal path, 



(5.14) 



(dp k /dk) > 0. 
opt. 
k=k* 



Notice, however, that on an optimal path it is not necessarily true that p, 
is monotonic in k. 

In the analysis of this section, we have so far used an important curva- 



ture assumption: The assumption that U"(y ) < 0, which implies that the 



-18- 

instantaneous preference map in (y T ,y r ) space is str ictly convex. In or- 
der to study the limiting behavior of our model, we will relax this assump- 
tion in what follows. 

Proposition 6 : If, in the preceding analysis [Equations (5.1) - 
(5.14)], we replace the assumption U"(y ) < with the assumption that 
U"(y_) = 0, then along the nonspecialized segment of the optimal trajectory , 
the price of capital p, is a constant, independent of the capital-labor 

ratio k, (dp /dk) = 0. 
opt. 

Proof: Maximization of national income implies from (5.4) that 



(5.15) q = Bp k , 

• 

where the constant 8= U' > 0. But from (5.5), q = only if k* is the 
unique root to (5.7). From (2.5), k * uniquely determines the consumption 
price of capital Pi*> which in turn uniquely determines the utility price 
of capital q*, from (5.15). If p, < Pv*» then by (2.8) and (5.5), q must 
fall at a rate faster than 6 + n. If p, > Pi.*» then q must rise at a rate 
faster than 6 + n. Therefore, in order for the condition (5.6) to hold, 
on an optimal (nonspecialized) trajectory p, must be constant and equal 
to (6/q*). 2 

6. Optimal Fiscal and Monetary Policy 

We return to the analysis of the mixed economy described in Sections 
2-4. The government is assumed to have two goals: (1) The maintenance 
of price stability; and (2) The constrained maximization of a utility 
functional based on the stream of per-capita consumption. At each moment, 



The equation U(y r ) + qy T = constant describes the relevant indifference 
curve . 

2 
A complete discussion of this model with linear criterion functional ap- 
pears in [9] and [14] where by convention 6=1. A fuller account of the 
nature of the "transversality condition" (5.6) appears in Shell [10]. 



-19- 

the government possesses two policy tools that it can employ in pursuit 
of these goals: (1) The composition of the government debt ( monetary 
policy ) which is reflected in the debt-money ratio x; and (2) The size 
of the per-capita government deficit d ( fiscal policy) . Government 
action is constrained by the behavior of producers (described in Sec- 
tion 2) , by the behavior of asset-holders (described in Section 3) , 
and by individuals' saving behavircr (described in Section 4). 

Formally, the government chooses time paths for x(t) and d(t) 
that maximize the welfare functional 

00 

(5.1) / U[y r (t)]e" 5t dt, 

o « 

subject to the policy constraint that p (t) = p for all t ^ 0. 
From Section 5, we know that on an optimal trajectory, the price of 
capital p, is a function only of the capital-labor ratio k. We can 
thus write equation (4.7) as 

• 

(6.1) k = yi [k, p k (k)] - nk. 

Since the government maintains a stable price level, from (4.1) and 
(4.4), the change in the per-capita government debt on an optimal path 
is given by 

(6.2) g = <J>[k, p k (k)] - ng. 

The dynamic behavior of the mixed economy with optimal monetary and fis- 
cal policy is completely described by the system (6.1) - (6.2). The 
capital-labor ratio k uniquely determines the optimal consumption price 
of capital p, (k) (as described in Section 5). At the given target p , 

K. in 

the government must choose x instantaneously so that the asset market 
equilibrium p is the same as the optimal p, (k). Given k and p, together, 



-20- 



producers determine the supplies of investment and consumption goods. The 

government then must adjust its deficit so that the market for consumption 

goods also clears at the given target p . 

m 

From Proposition 2, we know that the long-run optimal capital-labor 
ratio k* is unique. We also know by Proposition 3 that for k < k*, k > 0; 
for k > k*, k < 0. Thus in the phase diagrams of Figure III and Figure 
IV, k = only on a vertical line in the (g,k) plane. 

From (6.2), we know that g = if and only if 

(6.3) g = <|>[k, p k (k)]/n 

In order to find the slope of the curve described in (6.3), we totally 
differentiate (4.4) along an optimal trajectory , yielding 

, fi A^ ** -JL.!^ _s_^c/ r ^k) ^1 !£l/^k) foe) 
1 ; dk ~ 1-s 9k 1-s 3p k \dk>bpt. " p k 3k p k 3p k ydk/opt. y l\dkyopt. 

from the commodity market clearing equation (4.3). Combining (2.2) and (2.3) 
with Proposition 5 yields that the (dc}>/dk) may be either positive or nega- 
tive in the case with U"(y ) < 0. However, in the limiting case of the 
linear criterion functional where U"(y ) = 0, we can show that the optimal 
per-capita deficit <}>(•) is an increasing function of the capital-labor ratio. 
This is a consequence of the fact that in this case p, is constant on the 
optimal path, and so as k rises, the output of consumption goods rises faster 
than total income. The growing deficit is necessary to close this defla- 
tionary gap. 

Proposition 7 : In the case where the objective functional is linear 
in per-capita consumption, U" (y ) = 0, (d<{>/dk) > 0. 

Proof : Again, restricting our attention to cases of non-specializa- 
tion, Proposition 6 yields that (dp, /dk) =0. Since 



-21- 



* P m = Sy C " t 1 " 8 ^!' 



4 dk Lt. 



3y r 3y T 

'opt 



by (2.3). 

Also, since the right-hand sides of (6.1) and (6.2) are continuously 
differentiable, when the objective functional is "nearly" linear, the 
optimal per-capita deficit is an increasing function of the capital- 
labor ratio. 

On the assumption that (d<J)/dk) > 0, which implies that (dg/dk)._ n > 0, 
the system (6.1) - (6.2) is described in the phase diagram of Figure III. 
The stationary (balanced growth) solution to (6.1) - (6.2), (g*,k*), is 
unique. There is no reason, however, that g* must be positive. The 
question is whether the private sector will save too much or too little 
at the long-run optiaml income level y* to maintain the long-run optimal 
capital-labor ratio. If it saves too little, (sy*/p *) < nk*, the govern- 
ment will be forced to make up for this by running a surplus, and in the 
long run a surplus implies some indebtedness of the citizens to the gov- 
ernment. Both the surplus and the net indebtedness will grow in absolute 
value at the same rate as the population. If the community saves too 
much, (sy*/p *) > nk*, the government is forced to dissave constantly 
through a deficit that maintains a constant per-capita stock of debt. 

We do not need to restrict our attention to the case where (dcfi/dk) > 0. 
In Figure IV, we describe the behavior of the mixed economy for the case 
where (d$/dk) changes sign. In both cases (Figures III and IV), the balanced 
growth equilibrium (g*,k*) is globally stable. 

We now describe in detail the development of the mixed economy with 
optimal fiscal and monetary policy which is described by equations (6.1) 
and (6.2). 



-22- 




Figure III 



-23- 




Figure IV 



-24- 

Proposition 8 : (a) The balanced growth state (g*,k*) is unique and 

is globally stable. That is, the mixed economy with optimal fiscal and 

monetary policy tends to (g*,k*) independently of initial endowments 

(g ,k ). (b) On an optimal trajectory, k is either monotonicalfy decreasing 

or monotonically increasing. (c) If the criterion functional is linear, 

u "(y r ) = 0» or "nearly" linear, min U"(y r ) sufficiently close to zero so 

y C 
that (d<J)/dk) > 0, then (dg/dt) changes sign at most once. (d) In 

any case, however, x(t) tends to some long-run limit x*; and there exists 
a time T after which g(t) is monotonic. 

Proof : (a) Uniqueness follows from the fact that <£ is a single- 
valued function of k. From Proposition 3, k = ^(k), where in the neigh- 
borhood of k*, iK») is a decreasing function. Taking a linear approxima- 
tion to (6.1) - (6.2) about (g*,k*) yields the associated characteristic 

2 
equation x + [n - i/>'(k*)] x - ni^' (k*) = 0, where x is the characteristic 

root. Since the sum of the roots is negative, while the product of the 

roots is positive, (g*,k*) is locally stable and thus globally stable; 

(b) follows directly from Proposition 3; (c) follows from equation (6. A), 

Proposition 7, the continuous differentiability of (6.1) - (6.2), and 

from Figure III. (d) Since P,(t), g (t) , and k(t) tend to limits, x(t) 

also tends to a proper limit. We assume that the production functions are 

well behaved, and therefore the g = curve is well behaved — having a finite 

number of local extrema in any finite interval. Hence, there must exist 

£ > such that g = is monotonic in the region [k* - £,k*] and in the 

region [k* , k* + e]. Since an optimal trajectory spends all but a finite 

amount of time in one of these two regions, there must exist 0<T < °°, 

such that for t > T, g(t) is monotonic. 



-25- 

7. Burden of the Debt and Comparative Dynamics 

From Proposition 8, we conclude that in our model, there is no per se 
burden of the government debt . Not only is the long-run debt g* independent 
of iititial debt g , but also the entire trajectory of per-capita consumption 
(and thus welfare) is independent of the level of the debt that the economy 
inherits. The real consumption opportunity that is left to a generation is 
entirely described by its inherited capital-labor ratio and is unaffected 
by its inherited government indebtedness. 

The conclusion that there is no per se burden of the government debt 
does not depend upon whether or not the saving of the private sector de- 
pends upon the private sector's wealth. This conclusion, however, is 
crucially dependent upon the assumption made in Section 3 that asset de- 
mand functions H(«)» J(*)» and L(-) are sufficiently flexible so that holding 
the price level constant, the government by monetary and fiscal policy is 
able to achieve any efficient allocation of output regardless of the value 
of g. If such flexibility does not exist, then the analysis of optimal 
fiscal and monetary policy outlined in Section 6 would need to be altered. 
Without this flexibility in the asset market, the government would have to 
pursue a "second best" fiscal and monetary policy — achieving less welfare 
than is possible in the fully controlled economy. In this case, the gov- 
ernment would also have to consider the trade-off between relative price 
stability of the consumer price level and the current and future consump- 
tion opportunities of the economy. 

We now develop certain propositions in comparative dynamics for the 
mixed economy in which the government pursues the "first-best" optimal 
fiscal and monetary policy described in Section 6. 

Proposition 9 : Given technology, the long-run optimal capital-labor 
ratio k* depends solely upon the government's pure rate of time discount 6. 



-26- 

The more impatient the government (i.e., the higher 6), the lower the long- 
run optimal capital-labor ratio k*; (dk*/d6) < 0. 

Proof : The proof is based on Figure II. In the (q,k) plane, the k = 
schedule is independent of 6. However, from (5.5) and (5.7), we have along 
the q = schedule, the higher 5, the lower is k . From (2.5), it follows 
that on the q = schedule, (3p,/36) < 0. From (5.11) and (5.12), it follows 
that as 6 increases, the q = must shift to the southwest. The result 
follows immediately. 

Proposition 9 also applies to the limiting case of the linear cri- 
terion functional. In this case, non-specialized maximization of national 
income H requires that q = 3p, , where the constant 3 = U' > 0. Since 
(5.5) and (5.7) tell us that q = for the unique k* that solves fjOO = 
6 + n, we have that q = for a unique consumption price of capital p* and 
utility price of capital q*. Therefore, q = only on a horizontal line 
in the (q,k) plane of Figure II. Since in this case, U" = 0, (5.12) yields 
that (3o)/3k) = so the k = curve is independent of 6 and has a positive 
slope. As 6 increases, the q = line shifts to the south, while the k = 
curve does not shift. Thus, (3k*/36) < 0. 

Proposition 10 : The derivative (3g*/3<5) is either positive, negative, 
or zero depending upon the technology and the private sector's savings 
propensity s. 

Proof : Setting k = and differentiating in (4.8) yields 
3p k * n - (3 yi /3k) 



Since 



3k* (3y x /3p k ) 



* 

d(j>* 3$ it_ P k 
dk* 3k 3p k 3k* 



we have, from (4.5) and (4.6), that 



> 0. 



-27- 



d^. i_(^_)( d icY *i J_s_) !zc K ^iK 

dk* ~ p m \l-s/\9k / p k 3k \l-s/ 3p k dk* 3p k dk* 



dp, 

■ k 

y I dk* 






which is either positive, negative, or zero. That is 
(d(J)*/dk*) ^ as (3cp/3k) ^ - (3<|>/8p v ) (3p*/3k*) . 

Proposition 10 tells us that if we compare two economies with the same 
technology and the same individual savings behavior (equal s) , the economy 
whose government is less impatient (the government with the lower 5) may 
seek a higher long-run per-capita government debt g*. Thus, depending 
upon technology and individual savings behavior, the economy s eeking the 
higher long-run per-capita consumption y * (and thus the high er lon g-run 
capital-labor ratio k*) may follow a fiscal and monetary policy lead ing 
to a higher long-run per-capita debt g* . 

Proposition 10 contrasts sharply with the widely held belief that the 
larger the long-run debt, the lower is the long-run capital stock. We 
emphasize, however, that this proposition depends on partiuclar assumptions 
about savings and technology. 

Proposition 11 : Taking the government's objective functional as given, 
the higher the community's savings propensity s, the higher is the long-run 
debt g*, (3g*/3s) > 0. 

Proof : From (4.3), 

(l-s)<j>p° = y* - (l-s)y* 

where asterisks are used to indicate long-run equilibrium values of variables, 
Differentiating yields 

p° (3<}>*/3s) = y*/(l-s) 2 > 0. 

Til v> 



-28- 

The proposition follows immediately from (4.1). 

This leads to the natural classification of economies into those which 
are long-run "oversavers" and those which are long-run "undersavers." If, 
given the government's discount rate 6, s is sufficiently large, then long- 
run fiscal policy will lead the government to a net debtor position 
(g* > 0). If s is sufficiently small, then the government will become a 
net creditor (g* < 0) . By continuity for every 6 > 0, there must exist 
a < s < 1 such that g* = 0. 

The converse is not true. If s is sufficiently high, then a govern- 
ment policy in which lim g(t) = will lead to a capital-labor ratio for- 

ever bonded above the Golden Rule. We know by the Phelps -Koopmans Theorem 
[4], that such programs are dynamically inefficient. In such a case, ef- 
ficiency will require the government to be a long-run net debtor (g* > 0) . 
Thus, in this case, given the private sector's saving propensity s, there 
would be no rate of time preference 6 > consistent with a long-run zero 
debt, g* = 0. 1 






Indeed, there would be no dynamically efficient infinite program consis- 
tent with lim g(t) = 



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[1] Cass, Do, "Studies in the Theory of Optimal Economic 
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[3] Nelson, RoR u , "Full Employment Policy and Economic 
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[8] Rybczynski, ToMo, "Factor Endowments and Relative 

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[10] Shell, Ko , "Applications of Pontryagin's Maximum 

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[14] Uzawa, H„ , "Optimal Growth in a Two-Sector Model of 
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